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Search: a017233 -id:a017233
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A008585
a(n) = 3*n.
+10
296
0, 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, 39, 42, 45, 48, 51, 54, 57, 60, 63, 66, 69, 72, 75, 78, 81, 84, 87, 90, 93, 96, 99, 102, 105, 108, 111, 114, 117, 120, 123, 126, 129, 132, 135, 138, 141, 144, 147, 150, 153, 156, 159, 162, 165, 168, 171, 174, 177
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
COMMENTS
If n != 1 and n^2+2 is prime then n is a member of this sequence. - Cino Hilliard, Mar 19 2007
Multiples of 3. Positive members of this sequence are the third transversal numbers (or 3-transversal numbers): Numbers of the 3rd column of positive numbers in the square array of nonnegative and polygonal numbers A139600. Also, numbers of the 3rd column in the square array A057145. - Omar E. Pol, May 02 2008
Numbers n for which polynomial 27*x^6-2^n is factorizable. - Artur Jasinski, Nov 01 2008
1/7 in base-2 notation = 0.001001001... = 1/2^3 + 1/2^6 + 1/2^9 + ... - Gary W. Adamson, Jan 24 2009
A165330(a(n)) = 153 for n > 0; subsequence of A031179. - Reinhard Zumkeller, Sep 17 2009
A011655(a(n)) = 0. - Reinhard Zumkeller, Nov 30 2009
A215879(a(n)) = 0. - Reinhard Zumkeller, Dec 28 2012
Moser conjectured, and Newman proved, that the terms of this sequence are more likely to have an even number of 1s in binary than an odd number. The excess is an undulating multiple of n^(log 3/log 4). See also Coquet, who refines this result. - Charles R Greathouse IV, Jul 17 2013
Integer areas of medial triangles of integer-sided triangles.
Also integer subset of A188158(n)/4.
A medial triangle MNO is formed by joining the midpoints of the sides of a triangle ABC. The area of a medial triangle is A/4 where A is the area of the initial triangle ABC. - Michel Lagneau, Oct 28 2013
From Derek Orr, Nov 22 2014: (Start)
Let b(0) = 0, and b(n) = the number of distinct terms in the set of pairwise sums {b(0), ... b(n-1)} + {b(0), ... b(n-1)}. Then b(n+1) = a(n), for n > 0.
Example: b(1) = the number of distinct sums of {0} + {0}. The only possible sum is {0} so b(1) = 1. b(2) = the number of distinct sums of {0,1} + {0,1}. The possible sums are {0,1,2} so b(2) = 3. b(3) = the number of distinct sums of {0,1,3} + {0,1,3}. The possible sums are {0, 1, 2, 3, 4, 6} so b(3) = 6. This continues and one can see that b(n+1) = a(n).
(End)
Number of partitions of 6n into exactly 2 parts. - Colin Barker, Mar 23 2015
Partial sums are in A045943. - Guenther Schrack, May 18 2017
Number of edges in a maximal planar graph with n+2 vertices, n > 0 (see A008486 comments). - Jonathan Sondow, Mar 03 2018
REFERENCES
A. H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, p. 189.
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..5000
J. Coquet, A summation formula related to the binary digits, Inventiones Mathematicae 73 (1983), pp. 107-115.
Charles Cratty, Samuel Erickson, Frehiwet Negass, and Lara Pudwell, Pattern Avoidance in Double Lists, preprint, 2015.
A. S. Fraenkel, New games related to old and new sequences, INTEGERS, Electronic J. of Combinatorial Number Theory, Vol. 4, Paper G6, 2004.
John Graham-Cumming, The hollow triangular numbers are divisible by three (2013)
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 315
Tanya Khovanova, Recursive Sequences
D. J. Newman, On the number of binary digits in a multiple of three, Proc. Amer. Math. Soc. 21 (1969) 719-721.
Franck Ramaharo, Statistics on some classes of knot shadows, arXiv:1802.07701 [math.CO], 2018.
Luis Manuel Rivera, Integer sequences and k-commuting permutations, arXiv preprint arXiv:1406.3081 [math.CO], 2014-2015.
Wikipedia, Maximal planar graphs
Index entries for linear recurrences with constant coefficients, signature (2,-1).
FORMULA
G.f.: 3*x/(1-x)^2. - R. J. Mathar, Oct 23 2008
a(n) = A008486(n), n > 0. - R. J. Mathar, Oct 28 2008
G.f.: A(x) - 1, where A(x) is the g.f. of A008486. - Gennady Eremin, Feb 20 2021
a(n) = Sum_{k=0..inf} A030308(n,k)*A007283(k). - Philippe Deléham, Oct 17 2011
E.g.f.: 3*x*exp(x). - Ilya Gutkovskiy, May 18 2016
From Guenther Schrack, May 18 2017: (Start)
a(3*k) = a(a(k)) = A008591(n).
a(3*k+1) = a(a(k) + 1) = a(A016777(n)) = A017197(n).
a(3*k+2) = a(a(k) + 2) = a(A016789(n)) = A017233(n). (End)
EXAMPLE
G.f.: 3*x + 6*x^2 + 9*x^3 + 12*x^4 + 15*x^5 + 18*x^6 + 21*x^7 + ...
MATHEMATICA
Range[0, 500, 3] (* Vladimir Joseph Stephan Orlovsky, May 26 2011 *)
PROG
(Magma) [3*n: n in [0..60]]; // Vincenzo Librandi, Jul 23 2011
(Maxima) makelist(3*n, n, 0, 30); /* Martin Ettl, Nov 12 2012 */
(Haskell)
a008585 = (* 3)
a008585_list = iterate (+ 3) 0 -- Reinhard Zumkeller, Feb 19 2013
(PARI) a(n)=3*n \\ Charles R Greathouse IV, Jun 28 2013
CROSSREFS
Row / column 3 of A004247 and of A325820.
Cf. A016957, A057145, A139600, A139606, A001651 (complement), A032031 (partial products), A190944 (binary), A061819 (base 4).
Cf. A031179, A008486, A008591, A017197, A017233, A045943.
KEYWORD
nonn,easy
AUTHOR
N. J. A. Sloane
EXTENSIONS
Partially edited by Joerg Arndt, Mar 11 2010
STATUS
approved
A016789
a(n) = 3*n + 2.
+10
196
2, 5, 8, 11, 14, 17, 20, 23, 26, 29, 32, 35, 38, 41, 44, 47, 50, 53, 56, 59, 62, 65, 68, 71, 74, 77, 80, 83, 86, 89, 92, 95, 98, 101, 104, 107, 110, 113, 116, 119, 122, 125, 128, 131, 134, 137, 140, 143, 146, 149, 152, 155, 158, 161, 164, 167, 170, 173, 176, 179
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,1
COMMENTS
Except for 1, n such that Sum_{k=1..n} (k mod 3)*binomial(n,k) is a power of 2. - Benoit Cloitre, Oct 17 2002
The sequence 0,0,2,0,0,5,0,0,8,... has a(n) = n*(1 + cos(2*Pi*n/3 + Pi/3) - sqrt(3)*sin(2*Pi*n + Pi/3))/3 and o.g.f. x^2(2+x^3)/(1-x^3)^2. - Paul Barry, Jan 28 2004 [Artur Jasinski, Dec 11 2007, remarks that this should read (3*n + 2)*(1 + cos(2*Pi*(3*n + 2)/3 + Pi/3) - sqrt(3)*sin(2*Pi*(3*n + 2)/3 + Pi/3))/3.]
Except for 2, exponents e such that x^e + x + 1 is reducible. - N. J. A. Sloane, Jul 19 2005
The trajectory of these numbers under iteration of sum of cubes of digits eventually turns out to be 371 or 407 (47 is the first of the second kind). - Avik Roy (avik_3.1416(AT)yahoo.co.in), Jan 19 2009
Union of A165334 and A165335. - Reinhard Zumkeller, Sep 17 2009
a(n) is the set of numbers congruent to {2,5,8} mod 9. - Gary Detlefs, Mar 07 2010
It appears that a(n) is the set of all values of y such that y^3 = k*n + 2 for integer k. - Gary Detlefs, Mar 08 2010
These numbers do not occur in A000217 (triangular numbers). - Arkadiusz Wesolowski, Jan 08 2012
A089911(2*a(n)) = 9. - Reinhard Zumkeller, Jul 05 2013
Also indices of even Bell numbers (A000110). - Enrique Pérez Herrero, Sep 10 2013
Central terms of the triangle A108872. - Reinhard Zumkeller, Oct 01 2014
A092942(a(n)) = 1 for n > 0. - Reinhard Zumkeller, Dec 13 2014
a(n-1), n >= 1, is also the complex dimension of the manifold E(S), the set of all second-order irreducible Fuchsian differential equations defined on P^1 = C U {oo}, having singular points at most in S = {a_1, ..., a_n, a_{n+1} = oo}, a subset of P^1. See the Iwasaki et al. reference, Proposition 2.1.3., p. 149. - Wolfdieter Lang, Apr 22 2016
Except for 2, exponents for which 1 + x^(n-1) + x^n is reducible. - Ron Knott, Sep 16 2016
The reciprocal sum of 8 distinct items from this sequence can be made equal to 1, with these terms: 2, 5, 8, 14, 20, 35, 41, 1640. - Jinyuan Wang, Nov 16 2018
There are no positive integers x, y, z such that 1/a(x) = 1/a(y) + 1/a(z). - Jinyuan Wang, Dec 31 2018
As a set of positive integers, it is the set sum S + S where S is the set of numbers in A016777. - Michael Somos, May 27 2019
Interleaving of A016933 and A016969. - Leo Tavares, Nov 16 2021
Prepended with {1}, these are the denominators of the elements of the 3x+1 semigroup, the numerators being A005408 prepended with {2}. See Applegate and Lagarias link for more information. - Paolo Xausa, Nov 20 2021
This is also the maximum number of moves starting with n + 1 dots in the game of Sprouts. - Douglas Boffey, Aug 01 2022 [See the Wikipedia link. - Wolfdieter Lang, Sep 29 2022]
a(n-2) is the maximum sum of the span (or L(2,1)-labeling number) of a graph of order n and its complement. The extremal graphs are stars and their complements. For example, K_{1,2} has span 3, and K_2 has span 2. Thus a(3-1) = 5. - Allan Bickle, Apr 20 2023
REFERENCES
K. Iwasaki, H. Kimura, S. Shimomura and M. Yoshida, From Gauss to Painlevé, Vieweg, 1991. p. 149.
Konrad Knopp, Theory and Application of Infinite Series, Dover, p. 269
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..10000
D. Applegate and J. C. Lagarias, The 3x+1 semigroup, Journal of Number Theory, Vol. 177, Issue 1, March 2006, pp. 146-159; see also the arXiv version, arXiv:math/0411140 [math.NT], 2004-2005.
H. Balakrishnan and N. Deo, Parallel algorithm for radiocoloring a graph, Congr. Numer. 160 (2003), 193-204.
Allan Bickle, Extremal Decompositions for Nordhaus-Gaddum Theorems, Discrete Math, 346 7 (2023), 113392.
L. Euler, Observatio de summis divisorum p. 9.
L. Euler, An observation on the sums of divisors, arXiv:math/0411587 [math.HO], 2004-2009, p. 9.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 937
L. B. W. Jolley, Summation of Series, Dover, 1961, p. 16
Tanya Khovanova, Recursive Sequences
Konrad Knopp, Theorie und Anwendung der unendlichen Reihen, Berlin, J. Springer, 1922. (Original German edition of "Theory and Application of Infinite Series")
Fabian S. Reid, The Visual Pattern in the Collatz Conjecture and Proof of No Non-Trivial Cycles, arXiv:2105.07955 [math.GM], 2021.
Luis Manuel Rivera, Integer sequences and k-commuting permutations, arXiv preprint arXiv:1406.3081 [math.CO], 2014-2015.
Leo Tavares, Illustration: Capped Triangular Frames
Wikipedia, Sprouts (game)
Index entries for linear recurrences with constant coefficients, signature (2,-1).
FORMULA
G.f.: (2+x)/(1-x)^2.
a(n) = 3 + a(n-1).
a(n) = 1 + A016777(n).
a(n) = A124388(n)/9.
a(n) = A125199(n+1,1). - Reinhard Zumkeller, Nov 24 2006
Sum_{n>=1} (-1)^n/a(n) = (1/3)*(Pi/sqrt(3) - log(2)). - Benoit Cloitre, Apr 05 2002
1/2 - 1/5 + 1/8 - 1/11 + ... = (1/3)*(Pi/sqrt(3) - log 2). [Jolley] - Gary W. Adamson, Dec 16 2006
Sum_{n>=0} 1/(a(2*n)*a(2*n+1)) = (Pi/sqrt(3) - log 2)/9 = 0.12451569... (see A196548). [Jolley p. 48 eq (263)]
a(n) = 2*a(n-1) - a(n-2); a(0)=2, a(1)=5. - Philippe Deléham, Nov 03 2008
a(n) = 6*n - a(n-1) + 1 with a(0)=2. - Vincenzo Librandi, Aug 25 2010
Conjecture: a(n) = n XOR A005351(n+1) XOR A005352(n+1). - Gilian Breysens, Jul 21 2017
E.g.f.: (2 + 3*x)*exp(x). - G. C. Greubel, Nov 02 2018
a(n) = A005449(n+1) - A005449(n). - Jinyuan Wang, Feb 03 2019
a(n) = -A016777(-1-n) for all n in Z. - Michael Somos, May 27 2019
a(n) = A007310(n+1) + (1 - n mod 2). - Walt Rorie-Baety, Sep 13 2021
a(n) = A000096(n+1) - A000217(n-1). See Capped Triangular Frames illustration. - Leo Tavares, Oct 05 2021
EXAMPLE
G.f. = 2 + 5*x + 8*x^2 + 11*x^3 + 14*x^4 + 17*x^5 + 20*x^6 + ... - Michael Somos, May 27 2019
MAPLE
seq(3*n+2, n = 0 .. 50); # Matt C. Anderson, May 18 2017
MATHEMATICA
Range[2, 500, 3] (* Vladimir Joseph Stephan Orlovsky, May 26 2011 *)
LinearRecurrence[{2, -1}, {2, 5}, 70] (* Harvey P. Dale, Aug 11 2021 *)
PROG
(Haskell)
a016789 = (+ 2) . (* 3) -- Reinhard Zumkeller, Jul 05 2013
(PARI) vector(100, n, 3*n-1) \\ Derek Orr, Apr 13 2015
(Magma) [3*n+2: n in [0..80]]; // Vincenzo Librandi, Apr 14 2015
(GAP) List([0..70], n->3*n+2); # Muniru A Asiru, Nov 02 2018
(Python) for n in range(0, 100): print(3*n+2, end=', ') # Stefano Spezia, Nov 21 2018
CROSSREFS
First differences of A005449.
Cf. A002939, A017041, A017485, A125202, A017233, A179896, A017617, A016957, A008544 (partial products), A032766, A016777, A124388, A005351.
Cf. A087370.
Cf. similar sequences with closed form (2*k-1)*n+k listed in A269044.
Cf. A000096, A000217.
Cf. A016933, A016969.
KEYWORD
nonn,easy
AUTHOR
N. J. A. Sloane
STATUS
approved
A017173
a(n) = 9*n + 1.
+10
43
1, 10, 19, 28, 37, 46, 55, 64, 73, 82, 91, 100, 109, 118, 127, 136, 145, 154, 163, 172, 181, 190, 199, 208, 217, 226, 235, 244, 253, 262, 271, 280, 289, 298, 307, 316, 325, 334, 343, 352, 361, 370, 379, 388, 397, 406, 415, 424, 433, 442, 451, 460, 469, 478
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
COMMENTS
Also all the numbers with digital root 1; A010888(a(n)) = 1. - Rick L. Shepherd, Jan 12 2009
A116371(a(n)) = A156144(a(n)); positions where records occur in A156144: A156145(n+1) = A156144(a(n)). - Reinhard Zumkeller, Feb 05 2009
If A=[A147296] 9*n^2+2*n (n>0, 11, 40, 87, ...); Y=[A010701] 3 (3, 3, 3, ...); X=[A017173] 9*n+1 (n>0, 10, 19, 28, ...), we have, for all terms, Pell's equation X^2 - A*Y^2 = 1. Example: 10^2 - 11*3^2 = 1; 19^2 - 40*3^2 = 1; 28^2 - 87*3^2 = 1. - Vincenzo Librandi, Aug 01 2010
LINKS
Mohammed Yaseen, Table of n, a(n) for n = 0..10000
Tanya Khovanova, Recursive Sequences
Index entries for linear recurrences with constant coefficients, signature (2,-1).
FORMULA
G.f.: (1 + 8*x)/(1 - x)^2.
a(n) = 2*a(n-1) - a(n-2) with a(0)=1, a(1)=10. - Vincenzo Librandi, Aug 01 2010
E.g.f.: exp(x)*(1 + 9*x). - Stefano Spezia, Apr 20 2023
MATHEMATICA
Range[1, 1000, 9] (* Vladimir Joseph Stephan Orlovsky, May 28 2011 *)
LinearRecurrence[{2, -1}, {1, 10}, 60] (* Harvey P. Dale, Dec 27 2014 *)
PROG
(Sage) [i+1 for i in range(480) if gcd(i, 9) == 9] # Zerinvary Lajos, May 20 2009
(PARI) forstep(n=1, 500, 9, print1(n", ")) \\ Charles R Greathouse IV, May 28 2011
(Haskell)
a017173 = (+ 1) . (* 9)
a017173_list = [1, 10 ..] -- Reinhard Zumkeller, Feb 04 2014
CROSSREFS
Cf. A093644 ((9, 1) Pascal, column m=1).
Cf. A010888.
Cf. A116371, A156144, A156145.
Cf. A010701, A147296.
Numbers with digital root m: this sequence (m=1), A017185 (m=2), A017197 (m=3), A017209 (m=4), A017221 (m=5), A017233 (m=6), A017245 (m=7), A017257 (m=8), A008591 (m=9).
KEYWORD
nonn,easy
AUTHOR
N. J. A. Sloane
STATUS
approved
A017245
a(n) = 9*n + 7.
+10
20
7, 16, 25, 34, 43, 52, 61, 70, 79, 88, 97, 106, 115, 124, 133, 142, 151, 160, 169, 178, 187, 196, 205, 214, 223, 232, 241, 250, 259, 268, 277, 286, 295, 304, 313, 322, 331, 340, 349, 358, 367, 376, 385, 394, 403, 412, 421, 430, 439, 448, 457, 466, 475, 484
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,1
COMMENTS
Numbers whose digital root is 7. - Halfdan Skjerning, Mar 15 2018
LINKS
Table of n, a(n) for n=0..53.
Tanya Khovanova, Recursive Sequences
J. Laroche & N. J. A. Sloane, Correspondence, 1977
Leo Tavares, Illustration: Triple Triangular Frames
Index entries for linear recurrences with constant coefficients, signature (2,-1).
FORMULA
a(n)^2 = A156676(n+1) + A017137(n). - Reinhard Zumkeller, Jul 13 2010
a(n) = 18*n - a(n-1) + 5, with a(0)=7. - Vincenzo Librandi, Dec 24 2010
G.f.: (7+2*x)/(1-x)^2. - Vincenzo Librandi, Apr 30 2015
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n>2. - Vincenzo Librandi, Apr 30 2015
MATHEMATICA
Range[7, 1000, 9] (* Vladimir Joseph Stephan Orlovsky, May 28 2011 *)
Table[9 n + 7, {n, 0, 70}] (* or *) CoefficientList[Series[(7 + 2 x)/(1 - x)^2, {x, 0, 60}], x] (* Vincenzo Librandi, Apr 30 2015 *)
LinearRecurrence[{2, -1}, {7, 16}, 60] (* Harvey P. Dale, Jul 30 2024 *)
PROG
(Magma) [9*n+7: n in [0..60]]; // Vincenzo Librandi, Apr 30 2015
(PARI) vector(100, n, 9*n-2) \\ Derek Orr, Apr 30 2015
CROSSREFS
Cf. A008591, A017233.
KEYWORD
nonn,easy
AUTHOR
N. J. A. Sloane
STATUS
approved
A051062
a(n) = 16*n + 8.
+10
17
8, 24, 40, 56, 72, 88, 104, 120, 136, 152, 168, 184, 200, 216, 232, 248, 264, 280, 296, 312, 328, 344, 360, 376, 392, 408, 424, 440, 456, 472, 488, 504, 520, 536, 552, 568, 584, 600, 616, 632, 648, 664, 680, 696, 712, 728, 744, 760, 776, 792, 808, 824, 840
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,1
COMMENTS
Apart from initial term(s), dimension of the space of weight 2n cuspidal newforms for Gamma_0(97).
n such that 32 is the largest power of 2 dividing A003629(k)^n-1 for any k. - Benoit Cloitre, Mar 23 2002
Continued fraction expansion of tanh(1/8). - Benoit Cloitre, Dec 17 2002
If Y and Z are 2-blocks of a (4n+1)-set X then a(n-1) is the number of 3-subsets of X intersecting both Y and Z. - Milan Janjic, Oct 28 2007
General form: (q*n+x)*q x=+1; q=2=A016825, q=3=A017197, q=4=A119413, ... x=-1; q=3=A017233, q=4=A098502, ... x=+2; q=4=A051062, ... - Vladimir Joseph Stephan Orlovsky, Feb 16 2009
a(n)*n+1 = (4n+1)^2 and a(n)*(n+1)+1 = (4n+3)^2 are both perfect squares. - Carmine Suriano, Jun 01 2014
For all positive integers n, there are infinitely many positive integers k such that k*n + 1 and k*(n+1) + 1 are both perfect squares. Except for 8, all the numbers of this sequence are the smallest integers k which are solutions for getting two perfect squares. Example: a(1) = 24 and 24 * 1 + 1 = 25 = 5^2, then 24 * (1+1) + 1 = 49 = 7^2. [Reference AMM] - Bernard Schott, Sep 24 2017
Numbers k such that 3^k + 1 is divisible by 17*193. - Bruno Berselli, Aug 22 2018
REFERENCES
Letter from Gary W. Adamson concerning Prouhet-Thue-Morse sequence, Nov 11 1999
LINKS
Table of n, a(n) for n=0..52.
Mihaly Bencze, Problem 11508, The American Mathematical Monthly, Vol. 117, N° 5, May 2010, p. 459.
Milan Janjic, Two Enumerative Functions.
Tanya Khovanova, Recursive Sequences.
William A. Stein, Dimensions of the spaces S_k^{new}(Gamma_0(N)).
William A. Stein, The modular forms database.
Index entries for linear recurrences with constant coefficients, signature (2,-1).
FORMULA
a(n) = A118413(n+1,4) for n>3. - Reinhard Zumkeller, Apr 27 2006
a(n) = 32*n - a(n-1) for n>0, a(0)=8. - Vincenzo Librandi, Aug 06 2010
A003484(a(n)) = 8; A209675(a(n)) = 9. - Reinhard Zumkeller, Mar 11 2012
A007814(a(n)) = 3; A037227(a(n)) = 7. - Reinhard Zumkeller, Jun 30 2012
a(-1 - n) = - a(n). - Michael Somos, Jun 02 2014
Sum_{n>=0} (-1)^n/a(n) = Pi/32 (A244978). - Amiram Eldar, Feb 28 2023
From Elmo R. Oliveira, Apr 16 2024: (Start)
G.f.: 8*(1+x)/(1-x)^2.
E.g.f.: 8*exp(x)*(1 + 2*x).
a(n) = 8*A005408(n) = A008598(n) + 8 = A139098(n+1) - A139098(n).
a(n) = 4*A016825(n) = 2*A017113(n) = 2*a(n-1) - a(n-2) for n >= 2. (End)
MAPLE
A051062:=n->16*n+8; seq(A051062(n), n=0..50); # Wesley Ivan Hurt, Jun 01 2014
MATHEMATICA
Range[8, 1000, 16] (* Vladimir Joseph Stephan Orlovsky, May 31 2011 *)
Table[16n+8, {n, 0, 50}] (* Wesley Ivan Hurt, Jun 01 2014 *)
LinearRecurrence[{2, -1}, {8, 24}, 60] (* or *) NestList[#+16&, 8, 60] (* Harvey P. Dale, Aug 18 2019 *)
PROG
(Magma) [16*n+8: n in [0..50]]; // Wesley Ivan Hurt, Jun 01 2014
(PARI) a(n)=16*n+8 \\ Charles R Greathouse IV, May 09 2016
CROSSREFS
Cf. A005408, A008598, A017113, A106839, A139098, A244978.
Cf. A003484, A007814, A037227, A118413, A209675.
Cf. A003629, A016825, A017197, A017233, A098502, A119413.
KEYWORD
nonn,easy
AUTHOR
N. J. A. Sloane, Gary W. Adamson, Dec 11 1999
STATUS
approved
A122709
a(0)=1; thereafter a(n) = 9*n - 3.
+10
5
1, 6, 15, 24, 33, 42, 51, 60, 69, 78, 87, 96, 105, 114, 123, 132, 141, 150, 159, 168, 177, 186, 195, 204, 213, 222, 231, 240, 249, 258, 267, 276, 285, 294, 303, 312, 321, 330, 339, 348, 357, 366, 375, 384, 393, 402, 411, 420, 429, 438, 447, 456, 465, 474, 483
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
COMMENTS
Self-convolution of A122553.
LINKS
Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,-1).
FORMULA
a(0)=1, a(n) = 9*n - 3 = A008591(n) - 3 for n > 0.
a(n) = 2*a(n-1) - a(n-2) for n > 2; a(0)=1, a(1)=6, a(2)=15.
a(n) = a(n-1) + 9 for n > 1; a(0)=1, a(1)=6.
G.f.: ((1 + 2*x)/(1 - x))^2.
Equals binomial transform of [1, 5, 4, -4, 4, -4, 4, ...]. - Gary W. Adamson, Dec 10 2007
a(n) = A017233(n-1) for n > 0. - Georg Fischer, Oct 21 2018
E.g.f.: exp(x)*(9*x - 3) + 4. - Stefano Spezia, Mar 07 2023
MAPLE
seq(coeff(series(((1+2*x)/(1-x))^2, x, n+1), x, n), n = 0 .. 60); # Muniru A Asiru, Oct 21 2018
MATHEMATICA
Join[{1}, LinearRecurrence[{2, -1}, {6, 15}, 60]] (* Harvey P. Dale, Jun 12 2012 *)
PROG
(PARI) a(n)=max(9*n-3, 1) \\ Charles R Greathouse IV, Jan 17 2012
(PARI) Vec((1 + 2*x)^2 / (1 - x)^2 + O(x^100)) \\ Colin Barker, Jan 22 2018
(GAP) a:=[6, 15];; for n in [3..60] do a[n]:=2*a[n-1]-a[n-2]; od; Concatenation([1], a); # Muniru A Asiru, Oct 21 2018
CROSSREFS
Cf. A017233 (9n+6), A008591, A122553.
KEYWORD
nonn,easy
AUTHOR
Philippe Deléham, Sep 23 2006
EXTENSIONS
Edited by N. J. A. Sloane, Jan 23 2018
STATUS
approved
A017234
a(n) = (9*n + 6)^2.
+10
2
36, 225, 576, 1089, 1764, 2601, 3600, 4761, 6084, 7569, 9216, 11025, 12996, 15129, 17424, 19881, 22500, 25281, 28224, 31329, 34596, 38025, 41616, 45369, 49284, 53361, 57600, 62001, 66564, 71289
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,1
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).
FORMULA
From R. J. Mathar, Mar 21 2016: (Start)
G.f.: ( -36 - 117*x - 9*x^2 ) / (x-1)^3.
a(n) = 9*A016790(n). (End)
PROG
(Magma) [(9*n+6)^2: n in [0..35]]; // Vincenzo Librandi, Jul 25 2011
(PARI) a(n) = (9*n+6)^2; \\ Altug Alkan, Mar 21 2016
CROSSREFS
Cf. A000290 (n^2), A017233 (9*n+6).
KEYWORD
nonn,easy
AUTHOR
N. J. A. Sloane
STATUS
approved
A017235
a(n) = (9*n + 6)^3.
+10
1
216, 3375, 13824, 35937, 74088, 132651, 216000, 328509, 474552, 658503, 884736, 1157625, 1481544, 1860867, 2299968, 2803221, 3375000, 4019679, 4741632, 5545233, 6434856, 7414875, 8489664
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,1
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (4, -6, 4, -1).
FORMULA
G.f.: 27*(8 + 93*x + 60*x^2 + x^3)/(x-1)^4. - R. J. Mathar, Mar 20 2018
MATHEMATICA
(9*Range[0, 30]+6)^3 (* or *) LinearRecurrence[{4, -6, 4, -1}, {216, 3375, 13824, 35937}, 30] (* Harvey P. Dale, Feb 14 2018 *)
PROG
(Magma) [(9*n+6)^3: n in [0..35]]; // Vincenzo Librandi, Jul 25 2011
CROSSREFS
Cf. A000578 (n^3), A017233 (9*n+6).
KEYWORD
nonn,easy
AUTHOR
N. J. A. Sloane
STATUS
approved
A017239
a(n) = (9*n + 6)^7.
+10
1
279936, 170859375, 4586471424, 42618442977, 230539333248, 897410677851, 2799360000000, 7446353252589, 17565568854912, 37725479487783, 75144747810816, 140710042265625, 250226879128704, 425927596977747
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,1
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (8, -28, 56, -70, 56, -28, 8, -1).
FORMULA
a(n) = 8*a(n-1) - 28*a(n-2) + 56*a(n-3) - 70*a(n-4) + 56*a(n-5) - 28*a(n-6) + 8*a(n-7) - a(n-8); a(0)=279936, a(1)=170859375, a(2)=4586471424, a(3)=42618442977, a(4)=230539333248, a(5)=897410677851, a(6)=2799360000000, a(7)=7446353252589. - Harvey P. Dale, Feb 11 2015
MATHEMATICA
(9*Range[0, 20]+6)^7 (* or *) LinearRecurrence[{8, -28, 56, -70, 56, -28, 8, -1}, {279936, 170859375, 4586471424, 42618442977, 230539333248, 897410677851, 2799360000000, 7446353252589}, 20] (* Harvey P. Dale, Feb 11 2015 *)
PROG
(Magma) [(9*n+6)^7: n in [0..25]]; // Vincenzo Librandi, Jul 25 2011
CROSSREFS
Cf. A001015 (n^7), A017233 (9*n+6).
KEYWORD
nonn,easy
AUTHOR
N. J. A. Sloane
STATUS
approved
A269416
Expansion of 3*(2 - x)/((1 - x)*(1 + x)^2).
+10
1
6, -9, 15, -18, 24, -27, 33, -36, 42, -45, 51, -54, 60, -63, 69, -72, 78, -81, 87, -90, 96, -99, 105, -108, 114, -117, 123, -126, 132, -135, 141, -144, 150, -153, 159, -162, 168, -171, 177, -180, 186, -189, 195, -198, 204, -207, 213, -216, 222, -225, 231, -234, 240
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,1
COMMENTS
Alternating sum of A017233.
LINKS
Table of n, a(n) for n=0..52.
Index entries for linear recurrences with constant coefficients, signature (-1,1,1).
FORMULA
G.f.: 3*(2 - x)/((1 - x)*(1 + x)^2).
a(n) = -a(n-1) + a(n-2) + a(n-3).
a(n) = Sum_{k=0..n} (-1)^k*3*(3*k + 2).
a(n) = 3*((-1)^n*6*n + (-1)^n*7 + 1)/4.
Sum_{n>=0} 1/a(n) = log(3)/6 - Pi/(18*sqrt(3)) = 0.082335416765006179088425414... . - Vaclav Kotesovec, Feb 25 2016
a(n) = 3*(-1)^n*A007494(n+1). - R. J. Mathar, Jun 07 2016
EXAMPLE
a(0) = 1 + 2 + 3 = 6;
a(1) = 1 + 2 + 3 - 4 - 5 - 6 = -9;
a(2) = 1 + 2 + 3 - 4 - 5 - 6 + 7 + 8 + 9 = 15;
a(3) = 1 + 2 + 3 - 4 - 5 - 6 + 7 + 8 + 9 - 10 - 11 - 12 = -18;
a(4) = 1 + 2 + 3 - 4 - 5 - 6 + 7 + 8 + 9 - 10 - 11 - 12 + 13 + 14 + 15 = 24, etc.
MATHEMATICA
LinearRecurrence[{-1, 1, 1}, {6, -9, 15}, 53]
Table[3 ((6 (-1)^n n + 7 (-1)^n + 1)/4), {n, 0, 52}]
CROSSREFS
Cf. A000027, A017233.
KEYWORD
sign,easy
AUTHOR
Ilya Gutkovskiy, Feb 25 2016
STATUS
approved
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